Integration: the Area Problem

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The formula to find areas underneath curves derived from scratch

Tip: This isn't the place to ask a question because the teacher can't reply.

Key Questions

  • Answer:

    See below.

    Explanation:

    Question:
    When using integration to find an area, exactly which "area" is found?

    Since there are many different "area" problems that one will encounter in Calculus, taking a look at the type one usually meets early on, when studying the definite integral will help. Once there is some insight on this level, the area problems that will be presented as one advances through the study of Calculus will be easier to understand -- with this as background information.

    A typical type of area problem one meets early on has this form:
    # int_a^b f(x) dx #

    One can describe the integral as a calculation of the area under a curve - between the graph of the function, f(x), and the x-axis, bounded on the left by the vertical line # x = a #, and on the right by the line # x = b #.

    As the saying goes, "a picture is worth a thousand words". The blue colored area on the graph represents the area one would find by evaluating the integral in the equation above, with a = 2, and b = 5.

    enter image source here

    I hope that this presentation is helpful.

  • Let us look at the definition of a definite integral below.

    Definite Integral
    #\int_a^b f(x) dx=lim_{n to infty}sum_{i=1}^n f(a+iDelta x)Delta x#,
    where #Delta x ={b-a}/n#.

    If #f(x)ge0#, then the definition essentially is the limit of the sum of the areas of approximating rectangles, so, by design, the definite integral represents the area of the region under the graph of #f(x)# above the x-axis.

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